Download ppt

Mesh connected networks.
Sorting algorithms
Efficient Parallel Algorithms
COMP308
Lecture 14

Any nontrivial problem has no NC-algorithm on
the mesh, since the diameter is too large. The
problem of size n = k*k can be solved in time at
least 2k-1, since this is the number of steps for
communication between opposite corners
 For example we cannot compute on a mesh
minimum of n numbers in polylogarithmic time
since the communication between opposite
corners of the mesh takes more time than that
Sorting on the 1-dimensional
mesh

Algorithms on meshes can be considered as
"systolic" algorithms, systolic means very regular
and well synchronized
 Sorting on the 1-dimensional mesh k phases,
– in the first phase compare-exchange
(1,2) (3,4), (5,6), . . .
– In the second phase compare-exchange
(2,3) (4,5) (6,7) ... etc.
Sorting on 2-dim mesh

The first attempt to sort 2-dim mesh is to
sort all rows simultaneously then sort all
columns simultaneously
 However the whole table is not sorted in the
sense that elements of lower rows are all
larger than elements of upper rows.
Shearsort Algorithm
Correctness of Shearsort

Applying 0-1 principle, consider only 0-1
sequences
 Define the row to be dirty iff it contains at
least one 0 and at least on 1, other rows are
called clean
Main property
Correctness of Shearsort

Consider two adjacent dirty rows, assume w.l.o.g that the
first is an odd row the second an even row after sorting
these 2 rows according to their order we have the
following 3 possibilities

Start sorting columns by first comparing-exchanging
between of dirty rows decreases by half (one row per each
pair) these 2 rows, then one of them becomes clean, so the
number. Then these two rows will become (3 cases)

independently how we sort columns later the
number of dirty rows does not increase, the final
result of sorting columns does not depend of the
way of sorting them, the result always the same so
we could started with making exchanges between
these 2 rows.